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Chapter 30 | Atomic Physics 1347
Figure 30.20 Energy-level diagram for hydrogen showing the Lyman, Balmer, and Paschen series of transitions. The orbital energies are calculated using the above equation, first derived by Bohr.
Electron total energies are negative, since the electron is bound to the nucleus, analogous to being in a hole without enough kinetic energy to escape. As approaches infinity, the total energy becomes zero. This corresponds to a free electron with no
kinetic energy, since gets very large for large , and the electric potential energy thus becomes zero. Thus, 13.6 eV is
needed to ionize hydrogen (to go from –13.6 eV to 0, or unbound), an experimentally verified number. Given more energy, the electron becomes unbound with some kinetic energy. For example, giving 15.0 eV to an electron in the ground state of hydrogen strips it from the atom and leaves it with 1.4 eV of kinetic energy.
Finally, let us consider the energy of a photon emitted in a downward transition, given by the equation to be
Substituting , we see that
(30.29)
(30.30)
(30.31)
(30.32)
Dividing both sides of this equation by gives an expression for :
It can be shown that
is the Rydberg constant. Thus, we have used Bohr’s assumptions to derive the formula first proposed by Balmer years earlier as a recipe to fit experimental data.
